Closed immersion
In algebraic geometry, a closed immersion of schemes is a morphism of schemes f : Z → X {\displaystyle f:Z\to X} that identifies Z as a closed subset of X such that locally, regular functions on Z can be extended to X. The latter condition can be formalized by saying that f # : O X → f ∗ O Z {\displaystyle f^{\#}:{\mathcal {O}}_{X}\rightarrow f_{\ast }{\mathcal {O}}_{Z}} is surjective. An example is the inclusion map Spec ( R / I ) → Spec ( R ) {\displaystyle \operatorname {Spec} (R/I)\to \operatorname {Spec} (R)} of affine schemes induced by the canonical ring map R → R / I {\displaystyle R\to R/I} .